# -Prove square root of 5 is IRRational using {Abstract Algebra}

• May 25th 2009, 07:08 PM
MollyMATH
-Prove square root of 5 is IRRational using {Abstract Algebra}
Hey fellow mathletes, need help! (Headbang) Do not reply using anything but abstract!!
-Prove that the square root of 5 is irrational using abstract algebra; it is fairly urgent so anything will help, thanks!
• May 25th 2009, 11:21 PM
NonCommAlg
Quote:

Originally Posted by MollyMATH

Hey fellow mathletes, need help! (Headbang) Do not reply using anything but abstract!!

-Prove that the square root of 5 is irrational using abstract algebra;

it is fairly urgent so anything will help, thanks!

this is really an overkill proof: suppose $\sqrt{5} \in \mathbb{Q}.$ then $[\mathbb{Q}(\sqrt{5}):\mathbb{Q}]=1.$ but, by Eisenstein's criterion, $p(x)=x^2-5$ is irreducible over $\mathbb{Q}$ and thus $[\mathbb{Q}(\sqrt{5}):\mathbb{Q}]=2.$ contradiction!
• May 26th 2009, 09:09 AM
ThePerfectHacker
Quote:

Originally Posted by NonCommAlg
this is really an overkill proof: suppose $\sqrt{5} \in \mathbb{Q}.$ then $[\mathbb{Q}(\sqrt{5}):\mathbb{Q}]=1.$ but, by Eisenstein's criterion, $p(x)=x^2-5$ is irreducible over $\mathbb{Q}$ and thus $[\mathbb{Q}(\sqrt{5}):\mathbb{Q}]=2.$ contradiction!

Haha, that just gave me an idea. You know in math we try to find more and more elementary proofs. Well, how about we do the opposite for a change. Prove that $\sqrt{5}$ is irrational using the most overblown results from math that you can think of, or prove irrationality using a very long and confusing proof.